EPJ Web of Conferences 113, 0 4 0 0 8 (2016 )
DOI: 10.1051/epjconf/ 2016 113 040 08
C Owned by the authors, published by EDP Sciences, 2016
Few-Nucleon Research at TUNL: Probing Two- and ThreeNucleon Interactions with Neutrons
C.R. Howell1 , a , W. Tornow1 , b , and H. Witała2 , c
1
2
Department of Physics, Duke University and Triangle Universities Nuclear Laboratory, Durham, NC, USA
M. Smoluchowski Institute of Physics, Jagiellonian University, Cracow, Poland
Abstract. The central goal of few-nucleon research at the Triangle Universities Nuclear
Laboratory (TUNL) is to perform measurements that contribute to advancing ab-initio
calculations of nuclear structure and reactions. The program aims include evaluating
theoretical treatments of few-nucleon reaction dynamics through strategically comparing theory predictions to data, determining properties of the neutron-neutron interaction
that are not accessible in two-nucleon reactions, and searching for evidence of longrange features of three-nucleon interactions, e.g., spin and isospin dependence. This
paper will review studies of three- and four-nucleon systems at TUNL conducted using
unpolarized and polarized neutron beams. Measurements of neutron-induced reactions
performed by groups at TUNL over the last six years are described in comparison with
theory predictions. The results are discussed in the context of the program goals stated
above. Measurements of vector analyzing powers for elastic scattering in A=3 and A=4
systems, differential cross sections for neutron-deuteron elastic scattering and neutrondeuteron breakup in several final-state configurations are described. The findings from
these studies and plans for the coming three years are presented in the context of worldwide activities in this front, in particular, research presented in this session.
1 Introduction
The theoretical treatment of few-nucleon systems and light nuclei using realistic models of interactions between nucleons provides a formalism bridge for connecting phenomena in nuclear structure
and reactions to quantum chromodynamics (QCD). Studies of bound-state properties and reaction dynamics in these systems allow for assessments of two- and three-nucleon force models over a wide
range of phase space. The theory tool kit includes rigorous few-nucleon calculations, ab-initio structure calculations of light nuclei, phenomenological nuclear potential models, effective field theory
calculations and lattice QCD computations. Studies of three- and four-nucleon systems at TUNL are
discussed with emphasis on measurements sensitive to long-range three-nucleon interactions and to
features of the nucleon-nucleon (NN) interaction that are not accessible using two-nucleon systems.
The mechanism by which the residual strong nuclear force is produced by the color interactions
in QCD is one of nature’s great mysteries. Because hadrons are color neutral the static strong force
a e-mail: [email protected]
b e-mail: [email protected]
c e-mail: [email protected]
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Figure 1. Schematic diagram for coherent theoretical
treatment of nuclear systems starting from high
energies where perturbative QCD can be applied
going to low-energy nuclear phenomena where
mean-field potential models are most efficient.
between nucleons is zero. Therefore, the strong nuclear force must be of a dynamical nature similar
to Van der Waals interactions between charge-neutral atoms or molecules. For example, the spatial
features of the NN potential are qualitatively similar to the London dispersive potential [1], which is
due to the induced polarization of the atomic electron cloud relative to the nucleus. In this analogy
the NN interaction is associated with the distortion of the pion cloud about the massive core. Qualitatively both potentials are attractive with effective ranges of distances several times the diameter of
the interacting bodies and are repulsive at distances where the objects overlap. The dynamical nature
of the nuclear force provides for rich phenomena and complexity that makes analysis of the systems
interesting and challenging.
Over the last decade substantial progress has been made on developing formalism and computational methods that contribute to having theoretically coherent descriptions of nuclear phenomena
with origins in QCD. An ultimate goal would be to describe nuclear matter over wide distance and
energy scales with a QCD Lagrangian. Achieving this aim will likely require work done over generations of scientists, similar to other grand challenges in science and technology. A schematic diagram
is shown in Fig. 1 of a plausible hierarchy of organizing the theoretical treatments of nuclear systems
spanning a variety of phenomena. The scheme starts at ultra high energies, where the most fundamental degrees of freedom are quarks and gluons, and progressively evolves in complexity. This
diagram is intended only to represent coarse features that should be included in a coherent picture
of strongly interacting matter. In this framework the unifying concept is the residual strong force
between nucleons. At the top of the diagram, mean-field potentials that describe nuclear structure
properties, collective motion of nuclei and nuclear reactions should be derived from residual strong
interactions between nucleons. Ab-initio calculations of the structure of light nuclei [2–5] and fewnucleon reaction dynamics [6] enable refinement of two-nucleon and multi-nucleon interactions using
effective degrees of freedom. Current theory tools for describing the strong nuclear force (two- and
three-nucleon interactions) include semi-empirical potential models, e.g., Refs. [7–11], effective field
theory formulations of two-nucleon (2N) and three-nucleon (3N) interactions [12, 13], and Lattice
QCD (LQCD) calculations of few-nucleon systems, e.g., see Refs. [14, 15]. Descriptions of the
collective properties of nucleons in terms of effective field theories, e.g., Ref. [16] and LQCD, e.g.,
Ref. [17], are steps toward bridging gaps between QCD and theoretical treatments of few-nucleon
systems.
In this paper we discuss the applications of few-nucleon systems to probe features of three-nucleon
interactions (3NIs) and to evaulate current formulations of the neutron-neutron (nn) 1 S0 interaction.
The general approach is to use ab-initio calculations of few-nucleon systems or light nuclei to search
for observables that are sensitive to 3N force (3NF) effects or to the 1 S0 nn interaction. Such theory
studies are useful for guiding experimental efforts. However, experiment planning should not be
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Figure 2. Diagrams of additional 3NI terms when
increasing the χEFT expansion from N2 LO to N3 LO.
This figure is from Ref. [13]. Expanding to N3 LO
results in increasing the number of interaction terms.
However, there are no additional LECs when adding
these higher order terms to the 3NI expansion. The
diagrams can be divided into five topologies with the
upper three lines contributing to the long-range
component of the force, and the lower two lines
providing short-range interactions.
strictly limited by them; there should be a purely exploratory component to provide opportunities for
discovering effects not predicted by current theory. The sections that follow provide an overview of
few-nucleon research at TUNL with focus on 3NF effects and the nn interaction.
2 Three-nucleon Interactions
There is extensive evidence illustrating the importance of 3NIs in ab-initio theoretical treatment of
nuclei. Examples include the triton-binding energy discrepancy [8, 10] and systematic studies of
the ground and excited states of light nuclei using Monte-Carlo Green function simulations [2] and
no-core shell model calculations [3–5]. Also, in the calculation of the nuclear equation of state, the
energy per nucleon as a function of density, is highly sensitive to the inclusion of 3NFs, especially at
high nuclear densities relevant to the interior of neutron stars [18]. The indications for 3NF effects in
low-energy scattering data are less definitive.
There are two types of 3NIs used in modern ab-initio nuclear calculations, semi-empirical potential, e.g., Refs. [10, 11] and effective field theory formulations, e.g., Refs. [12, 19–21]. The
lowest order in Chiral Effective Field Theory (χEFT) where 3NI occur is next-to-next-to-leading order (N2 LO). In this order there are two low-energy constants (LECs) associated with 3NI diagrams,
often referred to as cD and cE [21]. These parameters can be determined by simultaneously fitting,
e.g., the triton binding energy and 2 and . In this session, König [22] described calculations of the quartet proton-deuteron (pd) scattering length as potentially being another observable for constraining the
values of LECs in EFT interactions.
The additional non-vanishing 3NI diagrams added when increasing the expansion to next-to-nextto-next-to-leading order (N3 LO) are shown in Fig. 2. The terms in N3 LO 3NIs can be grouped according to their effective range. The features of the high-momentum components of nuclear wavefunctions
and high-density nuclear matter, e.g., the interior of neutron stars as mentioned above, are sensitive
to the short-range part of the 3NI. The long-range component of the 3NI, which is the theme of this
session, can be probed by studying low-momentum components of nuclear wavefunctions via lowenergy few-nucleon reactions. Three presentations in this session are examples of such investigations.
Sagara et al. [23] describe the energy dependence of the space-star (SST) in pd breakup at proton
energies below 20 MeV. They found that the ratio of the cross-section data to calculations is nearly
constant with the data being about 8% lower than theory. In contrast, as shown in Figs. 3 and 4,
neutron-deuteron (nd) data for the SST are about 30% larger than theory. The difference between the
data and theory decreases with increasing neutron energy. In Fig. 3 the cross section is shown as a
function of the arc-length S of the kinematical curve at incident nucleon energy of 13.0 MeV. The
cross-section curves are Faddeev calculations using the CD-Bonn [7] NN potential and the TucsonMelbourne 3N potential model [10]. The pd data [26] plotted with the star symbols are about 10%
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Cross Section (mb/sr MeV)
Figure 3. Cross sections for the space-star
configuration in nd breakup at an incident neutron
energy of 13.0 MeV. The average angle of the
detected neutrons are θn1 =θn2 =50.3◦ and φ12 = 120◦ .
This figure is from Ref. [24]. The data are nd breakup
from Ref. [24] (solid circles), nd breakup from
Ref. [25] (open circles) and pd breakup from
Ref. [26] (stars). The curves are predictions of ab
initio nd breakup calculation with 3NI. The dotted
curve is a point-geometry calculation, and the solid
curve is a simulation for the TUNL experiment [24].
Figure 4. Cross sections for the space-star
configuration in nd breakup as a function of an
incident neutron energy. This figure is from Ref. [27].
The data are nd breakup from Refs. [28–30]
(squares), [27] (inverted triangles), [25, 31]
(triangles), [32] (circles), and [33] (diamond).
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Neutron Beam Energy (MeV)
lower than the theoretical predictions; the findings reported by Sagara at this conference [23] are consistent with these results. Long-range 3NIs that are not included in current calculations is a possible
cause of the discrepancy between data and theory for the SST in nucleon-deuteron breakup.
Also in this session, calculations presented by Girlanda [34] illustrate that a solution to the longstanding "Ay puzzle" in 3N and 4N systems might be provided through the fitting flexibility enabled
by the additional LECs for 3NIs in N4 LO EFT interactions. Recent measurements made at TUNL
of the vector analyzing power (Ay ) for neutron-3 He elastic scattering [35] are shown in Fig. 5. As in
the case of the 3N system, theory underpredicts the value of Ay (θ) at the maximum for n3 He elastic
scattering. One approach to interpreting the Ay issue in the 4N system is through the lens of isospin.
Data for both mixed isospin (T=0,1) and pure isospin (T=1) states exist for proton scattering from
3
H and 3 He, respectively. However, only data for the mixed isospin state (n3 He) is available for
neutron scattering. Esterline et al. [35] found that the energy dependence of the fractional difference
between data and theory was similar for the mixed isospin (T = 0, 1) scattering systems of p3 H and
n3 He and has a slope in the energy range studied opposite to that for the pure T=1 p3 He scattering.
Measurements of Ay (θ) for n3 H elastic scattering are planned using the pulsed polarized neutron beam
at TUNL. The goal of this experiment is to provide data that will complement the p3 He (T=1) data.
3 Studies of the Neutron-neutron Interaction at TUNL
In the two-nucleon system the values for the 1 S0 NN scattering length aNN and effective range parameter rNN are tightly constrained by limits on charge dependence and charge-symmetry violation in the
strong NN interaction. This feature along with the high sensitivity of these parameters to the strength
of the S-wave NN interaction provide means for probing few-nucleon systems and light nuclei for
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Figure 5. Ay (θ) for n-3 He elastic
scattering at several energies. This
figure is adapted from Ref. [35]. The
data are measurements from TUNL,
and the curves are ab-initio
calculations with the attributes
indicated in the legend.
3NF effects. For example, discrepancies in the values of ann from π− d capture data and those obtained
from some nd breakup measurements could indicate 3NF effects that are not included in current nd
breakup calculations [36] . A more recent observation of effects that could be due to 3NIs that are
not included in current nd calculations was reported by Witała and Glöckle in their analysis of nn
quasi-free scattering (QFS) cross-section data in nd breakup [37]. The resolution of the about 20%
discrepancy between data and theory using current 3NIs required substantial charge-symmetry breaking in the NN interactions beyond what is generally accepted; the value of rnn in their calculation was
about 12% larger than allowed by charge symmetry.
There are two experiments underway at TUNL to further investigate these issues in the nd continuum. One experiment is to measure the cross section for nn QFS scattering in nd breakup. This
experiment is being carried out at two beam energies using the pulsed neutron beam facility at TUNL.
Details and preliminary results for measurements at 10 MeV are presented by Malone et al. [38] in
this session. The other experiment is a measurement of the differential cross section for exclusive
photodisintegration of 3 H in the kinematic region of the nn FSI. These data will be used to determine a value of ann to investigate 3NF effects in tritium as probed by the photodisintegration reaction
at gamma-ray energies below 20 MeV. The conceptual design of the experiment is described in the
poster presentations by Friesen et al. [39] and Han et al. [40].
Acknowledgements
This work is supported in part by the U.S. Department of Energy under grant No. DE-FG02-97ER41033 and by
the Polish National Science Center under Grant No. DEC-2013/10/M/ST2/00420.
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